Polynomials and fractions

9000087503

Level:

Assuming $x\in \mathbb{R}\setminus \left \{-\frac{3} {2}\right \}$, find the quotient of the polynomials: \[ (x^{2} + x + 1) : (2x + 3) \]

$\frac{1} {2}x -\frac{1} {4} + \frac{\frac{7} {4} } {2x+3}$

$\frac{1} {2}x -\frac{1} {2} + \frac{\frac{7} {4} } {2x+3}$

$x + 2 + \frac{7} {2x+3}$

$x - 2 + \frac{7} {2x+3}$

9000087501

Level:

Assuming $x\in \mathbb{R}\setminus \left \{-1\right \}$, find the quotient of the polynomials. \[ (3x^{2} + 2x + 7) : (x + 1) \]

$3x - 1 + \frac{8} {x+1}$

$3x + 2 + \frac{8} {x+1}$

$3x - 1 - \frac{5} {x+1}$

$3x + 2 - \frac{5} {x+1}$

Suppose we are given the following equality of two fractions with nonzero denominators. From the given expressions, choose the one that by substituting to the starred position makes the equality true. \[ \frac{3 - 2x} {x - 2} = \frac{3(4x^{2} - 12x + 9)} {*} \]

$(3x - 6)(3 - 2x)$

$(x - 2)(2x - 3)$

$(x - 2)(9 - 4x)$

$(3x - 6)(2x - 3)$

9000088801

Level:

Find the domain of the following expression. \[ \frac{2x + 1} {6x^{2} + 3x} \]

$\mathbb{R}\setminus\left\{0,-\frac{1}{2}\right\}$

$\mathbb{R}\setminus\{0,- 2\}$

$\mathbb{R}\setminus\{0\}$

$\mathbb{R}\setminus\left\{0,\frac{1}{2}\right\}$

9000088802

Level:

Find the domain of the following expression. \[ \frac{a} {a^{2} + 9}\cdot \frac{a^{2} - 9} {a^{2} + 3a} \]

$\mathbb{R}\setminus\{0,- 3\}$

$\mathbb{R}\setminus\{3,- 3\}$

$\mathbb{R}\setminus\{0,3\}$

$\mathbb{R}\setminus\{- 3\}$

9000088804

Level:

Simplify the following expression. \[ \frac{2s - 8rs} {16r^{2} - 1} \]

$- \frac{2s} {4r+1}$

$\frac{2s} {4r+1}$

$\frac{2s} {4r-1}$

$\frac{2s} {1-4r}$

9000088805

Level:

Simplify the following expression. \[ \frac{a^{4} - 1} {1 - a^{2}} \]

$- a^{2} - 1$

$a^{2} + 1$

$a^{2} - 1$

$1 - a^{2}$

9000083601

Level:

Find the conditions on $x$ under which the expression \[ \frac{\frac{x-y} {x+y} -\frac{x+y} {x-y}} { \frac{xy} {x^{2}-y^{2}} } \] is well-defined.

$x\neq 0,\; y\neq 0,\; x\neq \pm y$

$x\neq - y$

$x\neq \pm y$

$x\neq 0,\; y\neq 0$

9000083602

Level:

Evaluate the following expression at $x = \frac{1} {2}$. \[ \frac{x^{2} - 2} {1 -\frac{1} {x}} \]

$\frac{7} {4}$

$-\frac{7} {4}$

$\frac{7} {2}$

$-\frac{7} {2}$

9000083605

Level:

Find the common denominator of the fractions. \[ \text{$ \frac{3x} {x^{2}+4x+4}$ and $ \frac{x+5} {x^{2}-4}$} \]

$(x + 2)^{2}(x - 2),\; x\neq \pm 2$

$(x + 2)(x - 4),\; x\neq \pm 2$

$(x + 2)^{2}(x - 4),\; x\neq \pm 2$

$(x + 2)(x - 4),\; x\neq \pm 2$

Polynomials and fractions

9000087503

9000087501

9000088807

9000088801

9000088802

9000088804

9000088805

9000083601

9000083602

9000083605

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